VAN DER VRIES. — MULTIPLE POINTS OP TWISTED CURVES. 505 



A curve C,« of order m 



liavinn; a point (or points) of multi- 

 plicity K 



can be cut out of a cone of order m 

 having Cm as base by a monoid 

 of order jW 



tliat has the multiple point (or 

 points) of Cm as point (or 

 points) of multiplicity . . ^' — 1 



B. 



1. Every curve C^ of order m can also be obtainefl as the partial 

 intersection of a cone of order m — 1 and a monoid of order fi, where 

 iu is to be determined. The cone Km—x is a cone that has an ordinarv 

 point of Cm as its vertex and Cm as its base. There being an infinite 

 number of ordinary points on C'^, the special positions of the vertex 

 from which every line drawn to C^ meets C,n in two or more points can 

 be avoided. The vertex can therefore be taken in such a way that the 

 cone of order m — 1 does not break up into cones of orders that are sub- 

 multiples of m — 1. Every edge of K,n \ thus has on it one point of 

 Cin in addition to that at the vertex ; Cm can therefore be cut out of 

 Km—\ by a monoid J^ . We shall consider the case of a curve that has 

 points of multiplicity k, at which the tangents lie on cones of order /• + 1 

 that have the lines from the multiple points to the vertex of /\'„_i as 

 ^•-tuple edges ; the case of curves with no multiple points being but a 

 special case of this. It can be shown, as in the previous case, tliat the 

 lines from the common vertex of A',„_i and M^ to these K-tuple points 

 are K-tuple lines on Km—\ and /.--tuple lines of kind III on M^ . The 



